geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Be?linson-Bernstein localization?
Representation theory is concerned with the study of algebraic structures via their representations. This concerns notably groups, directly or in their incarnation as group algebras, Hopf algebras or Lie algebras, and usually concerns linear representations, hence modules of these structures. But more generally representation theory also studies representations/modules/actions of generalizations of such structures, such as coalgebras via their comodules etc.
See also at geometric representation theory.
The fundamental concepts of representation theory have a particular natural formulation in homotopy theory and in fact in homotopy type theory, which also refines it from the study of representations of groups to that of ∞-representations of ∞-groups. This includes both discrete ∞-groups as well as geometric homotopy types such as smooth ∞-groups, the higher analog of Lie groups.
The key observation to this translation is that
an ∞-group $G$ is equivalently given by its delooping $\mathbf{B}G$ regarded with its canonical point (see at looping and delooping), hence the universal $G$-principal ∞-bundle
an ∞-action $\rho$ of $G$ on any geometric homotopy type $V$ is equivalently given by a homotopy fiber sequence of the form
hence by a $V$-fiber ∞-bundle over $\mathbf{B}G$ which is the $\rho$-associated ∞-bundle to the universal $G$-principal ∞-bundle (see at ∞-action for more on this).
Under this identification, the representation theory of $G$ is equivalently
the homotopy theory in the slice (∞,1)-topos over $\mathbf{B}G$;
the homotopy type theory in the context of/dependent on $\mathbf{B}G$.
More in detail, this yields the following identifications:
representation theory and equivariant cohomology in terms of (∞,1)-topos theory/homotopy type theory (FSS 12 I, exmp. 4.4):
homotopy type theory | representation theory |
---|---|
pointed connected context $\mathbf{B}G$ | ∞-group $G$ |
dependent type on $\mathbf{B}G$ | $G$-∞-action/∞-representation |
dependent sum along $\mathbf{B}G \to \ast$ | coinvariants/homotopy quotient |
context extension along $\mathbf{B}G \to \ast$ | trivial representation |
dependent product along $\mathbf{B}G \to \ast$ | homotopy invariants/∞-group cohomology |
dependent product of internal hom along $\mathbf{B}G \to \ast$ | equivariant cohomology |
dependent sum along $\mathbf{B}G \to \mathbf{B}H$ | induced representation |
context extension along $\mathbf{B}G \to \mathbf{B}H$ | restricted representation |
dependent product along $\mathbf{B}G \to \mathbf{B}H$ | coinduced representation |
spectrum object in context $\mathbf{B}G$ | spectrum with G-action (naive G-spectrum) |
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
character sheaf?, Harish Chandra transform
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Lecture notes:
Tammo tom Dieck, Representation theory, 2009 (pdf, pdf)
Constantin Teleman, Representation theory, lecture notes 2005 (pdf)
Joel Robbin, Real, Complex and Quaternionic representations, 2006 (pdf, pdf)
Textbook accounts
for finite groups:
Charles Curtis, Irving Reiner, Representation theory of finite groups and associative algebras, AMS 1962
Klaus Lux, Herbert Pahlings, Representations of groups – A computational approach, Cambridge University Press 2010 (author page, publisher page)
and more generally for compact Lie groups:
Tammo tom Dieck, Theodor Bröcker, Representations of compact Lie groups, Springer 1985 (doi:10.1007/978-3-662-12918-0)
William Fulton, Joe Harris, Representation Theory: a First Course, Springer, Berlin, 1991 (doi:10.1007/978-1-4612-0979-9)
Discussion via string diagrams/Penrose notation:
Jeffrey Ellis Mandula, Diagrammatic techniques in group theory, Southampton Univ. Phys. Dept. (1981) (cds:129911, pdf)
Predrag Cvitanović, Group Theory: Birdtracks, Lie’s, and Exceptional Groups, Princeton University Press July 2008 (PUP, birdtracks.eu, pdf)
(aimed at Lie theory and gauge theory)
Further references:
The relation to number theory and the Langlands program is discussed in